132 XIII INTEGRATION 5. Let A be Lebesgue measure on R, and let E be the topological vector space obtained by endowing the space ^(X) with the topology of simple convergence (12.15). Show that the linear form/V~»J/dA on E is not continuous. 6. Let (An) be a sequence of /x-integrable subsets of X, such that £ ft(Aw) < + 00. For n each integer k, let Gfc be the set of all x e X such that x e An for at least k values of n. Show that Gfc is integrable and that k • /x(G*) 7. Let X be a compact space, \L a positive measure on X, and (An) a sequence of integrable subsets of X such that inf />t(An) = m > 0. Show that the set B of points of X which n belong to infinitely many of the sets An is integrable, and that /x(B) ^ m. 8. Let A be Lebesgue measure on the interval I = [0, 1] and let Aw be a sequence of integrable sets in I such that inf A(A«) == m > 0. For each integer k, let $)k denote the set of 2k intervals of the form [j - 2~fc, (/ + I) • 2"k] in I (0 ^/ < 2*), and <g* the set of all the unions of sets of 25*. Show that there exists a decreasing sequence (10*20 of subsets of I such that I0 = I, I* e @*for all k, and a subsequence (Anfc) of the sequence (An) with the following properties: (i) A(Anr n (I - I*)) ^ JwA(I ~Ifc) for all r ^ k); (ii) A(A«r n J) ^ £wA(J) for all r ^ k and all J e 5D* contained in I* . (Proceed by in- duction on k, using the diagonal trick.) Deduce that there exists a subsequence (Brt) of (An) such that, for all k and all J e 3)fc contained in I* , the intersection of J with the sequence (B«) is not empty. (Use Problem 7 and the diagonal trick.) Deduce that p| Bn contains a non-empty compact set with no isolated points (and therefore n nondenumerable (Section 4.2, Problem 3(c))). 9. Let /be a real- valued function ^0 on X. Show that the mapping //, i— » ju,*(/) of M +(X) into R is continuous with respect to the vague topology (13.4) if and only iff is con- tinuous and compactly supported. The mapping /AI— >/x*(/) is lower semicontinuous with respect to the vague topology if and only if /is lower semicontinuous. 10. Let (/z,n) be an increasing sequence of positive measures on X. Suppose that the se- quence is bounded above in M+(X), and let ^ be its least upper bound (13.4.4). Show that, for every function/^ 0 on X, we have fi*(/) = lim /*?(/). n-»-oo 11. (a) Let (//.„) be a decreasing sequence of positive measures on X, and let /z be the greatest lower bound of the sequence in M+(X) (13.4.4). If /is a function ^>0 on X such that ju*(/) < + oo for all sufficiently large n, show that ^*(/) = lim /^(/). n-*oo (If g ^ 0 is lower semicontinuous and n*(g) < + oo, observe that there exists a se- quence (/zm) of continuous functions ^0 with compact supports, such that]£ hm <*g and p*(ff) «£ p*(hm) for alt n.) m (b) On the discrete space N, let //,„ be the measure such that /nn({w}) = 0 for m < n and 1 for m ^ n. Show that the greatest lower bound of the decreasing sequence (//.„) in M+(N) is 0, but that /z*(N) = + oo for all n. Fx. If K is the convex and vaguely compact set of positive measures of